Local nonlinear geometry and estimates · Part 8

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burst(x) =
q
2 betaF(x).

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Then gStep(β)−1 = √β ≤burst, and burst has rate 1/2. The bounded cutofffield used by the
chain estimates therefore has common envelope

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U(x) = burst(x) · fiery(x) · muRate(C, H),

of rate zero. This envelope is independent of and the Gaussian variable. Its nonnegativity and the mean bound supply the hypotheses of the pointwise chain estimate. If E has rate k1/2, then)| has rate k. The first mean-correction envelope is

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J(x) = dim(Color)U(x)b(x),

of rate k. Its caption is the parent of its fine bond. With support town contained in radius- (m+1)B and fine-link caption census, its sales price is at most

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ballCost((m + 1)B)L4|Axis| · J(x)
· exp(Asupport ballCost(m + 1)),

also of rate k. The higher quadratic-chain envelopes are capQ evaluated at U. Their series sales bound follows from the itinerary estimate with center coefficient proportional to

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1
2U2flagP 4Q2payE · scraps(B)

and ratio coefficient flagPQpayEscraps( The latter has every prescribed negative rate; the former has rate k for any prescribed k because U has rate zero. The itinerary summation consequently gives the chain sales bound of rate k. The support charge is its series index, and halo 4 is retained. The pointwise Gaussian chain and short-correction estimates supply capQ and J; this calculation retains their coupling and residual factors in the common budget.

For a finite caption family of density n with supports contained in radius-dB balls, a constant coefficient envelope has sales price at most ballCost(dB)n exp( ballCost()). Indeed a fixed site belongs to at most ballCost(dB)n supports, and each support has B-covering number at most ballCost(). This is the fixed-envelope counting used for the first correction and local Short letters.

True path supports and bundle geometry

The native support of a walk is the finite set of its visited sites, including its start and endpoint. Every visited site lies within its actual walk length of the start, by induction and the one-step separation bound. A walk of length at most M is therefore covered by one radius-M tile. For a longer walk, split offits first M steps. The two native supports meet at the split endpoint. The meeting-support covering inequality and induction give

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M coverNumberM(native(w)) ≤M + length(w), M > 0.

This uses the walk length with all repetitions and winding, even when the finite native set is much smaller.

A curvature slot’s actual support is the native support of its transport walk followed by the four-step plaquette. This equality follows by concatenating visit lists and wrapping their cover coordinates. Its slot length is exactly that concatenated length. A bundle of d slots rooted at x is the union of their supports with x inserted. All its supports meet at x, so the same covering inequality gives

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M coverNumberM(bundle) ≤(1 + d)M + totalLength.

Every bundle site also lies within totalLength of x, since each individual slot length is at most their nonnegative sum. For a new support Y contained in halo(parent()), the parent-distance inequality gives its extent from parent() at most (lB+1)(1+totalLength). Apply the coarse footprint bound with and the last bundle covering inequality. Define

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baseFee = preCost(l) + conversion(L, l)(1 + d),
legFee = conversion(L, l)/B,
originFee = lB + 1.

Then coverNumber The anchored-to-hit principle applies with degree equal to true totalLength, extent originFee, and those two covering coefficients. At each fixed fine root, its input budget is exactly the original slot-family length budget; the root delta and this budget form a product budget, so no volume factor is inserted beyond the parent population. This proves bundle, including the tail exp() for and its explicit payment. Atom footprints are exactly bundles of their slot legs; an ordered path pair is the two-slot bundle and has the sum of its two actual lengths.

Connected center traversal and halo tiling

The covering currency uses the periodic maximum-coordinate separation. A center set is connected at range b if any two centers can be joined through centers with successive separation at most b. A cover at radius M covers the support by the M halos of its centers and requires those centers to be connected at range 2M+1. Thus disconnected pieces of a support still incur the cost of connecting their cover; it is not an unrestricted ball-counting number.

There is a unit-speed line from x to y of duration separation(x,y): in each centered-lift coordinate of yx, truncate the displacement to the interval [j,j] at time j, then wrap and add x. Each coordinate changes by at most one per time step, and it stops at y once j reaches their maximum absolute displacement.

For a finite center set D connected at range b, build a traversal by adjoining one unvisited center at a time. Connectivity supplies an edge from the visited subset to its complement: take the first exit along any center path to an unvisited point. At the occurrence of its visited endpoint in the current traversal, insert the line to the new center and its reverse. The detour has length at most 2b and preserves all old visits. Starting with one center and repeating gives a unit-speed traversal visiting every center with length at most 2b|D|. The empty case has the zero-length trivial traversal. This proves the ring-path bound used in extent and collapse.

For tiling a radius-kB halo of one site, take the image of the integer grid [ scaled by B and translated to that site. Divide each centered coordinate of a halo point by B. Its quotient lies in that grid and its remainder has absolute value at most B, so one grid center covers it. The grid centers are connected through grid steps; wrapping may identify centers and can only reduce their number. Hence at most (2 centers suffice.

If X has a radius-mB cover with center set D, enlarge its balls to radius (m+1)B. Their original connecting steps are at most 2, so these enlarged halos meet along the center connections. Tile each enlarged ball by the preceding grid and join their connected covers at the meetings. This gives

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coverNumberB(X) ≤|D|[2(m + 1) + 1]4.

The same line construction proves that halos of centers connected at range 2M+1 are connected at unit range: the line between two centers lies in their two M halos, with the possible middle adjacent step included. These provide the connected-halo and tiling facts used by the collapse proof without requiring any injectivity of a periodic halo.

Coarse collapse of a connected covering

Write L for the block side. A fine site is within L/2 of its embedded parent. Embedded coarse-site separation is at least L times coarse separation, coordinatewise from centered lifts. The triangle inequality therefore gives

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Lseparation(parent(x), parent(y)) ≤separation(x, y) + L.

For a unit-step fine-site path, sample its parents every LB steps, 0. Consecutive sampled parents are within 2B by this inequality, and every parent along the path is within 2B of one sample. The set D of samples is connected at that range and satisfies

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LB|D| ≤N + LB.

This includes the final partial segment and allows repeated samples.

Realize the scale-M cover of a support X by a center set A of size coverNumber) whose union of M-balls is connected. Join two centers when their distance is at most 2M+1. This center graph is connected: a lattice path through the union passes from one ball to another only through overlapping or adjacent balls. A finite connected graph has a spanning tree, obtained by adding an edge from the already reached vertices to a new vertex until all vertices are reached. Traverse this tree out and back. Each tree edge is replaced by a lattice path of length at most 2M+1, and each is traversed twice, so the resulting unit-step path visits every center and has length at most + 2). The empty cover has the constant path and length zero. Join any two sites of X to covering centers by paths of length at most M. Their distance is at most |A|(4M+2)+2M. This proves the extent estimate, on the periodic lattice as well as on the integer cover.

Sample the center traversal at scale LB as just described. If, the parent of any support point is within 3B of its center’s parent, and that center’s parent is within 2B of D. Hence halo(parent()) is covered by the radius-(s+5)B halo of D, with its inherited connectedness. Tiling that halo at scale B costs at most |D|ballCost(s+6). Combining the sample count, and1 gives

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L coverNumberB(halo(sB)(parent(X)))
≤[L + 10 coverNumberM(X)] ballCost(s + 6).

For X empty both sides are immediate; otherwise its realized cover has at least one center, which pays the fixed endpoint term in this inequality. Dividing by L yields exactly the preCost and conversion constants used in the coarse hit budget. Thus all spatial block factors in that budget follow from parent separation, a connected cover traversal, and halo tiling.

Anchored budgets converted to hits

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The geometric counting principle is the following.
Suppose each fine anchor has budget
P
oi=y bieAdi ≤γ, the parent map has at most P fine sites per coarse site, and

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x ∈Xi ⇒separation(x, parent(oi)) ≤W(1 + di),
coverNumberB(Xi) ≤v + v′di.

Let 0, and assume 0 and . At fixed degree the coefficient with its new support weight is bounded by

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bi exp(Aj) · C(1 + j)k exp(av −2εj).

All relevant anchors lie above the coarse radius-W(1+j) ball about x. Their number is at most P ballCost(. Group a finite set of labels by anchor and apply the original anchored budget at each anchor. The resulting degree-j total is at most

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exp(av)γCP ballCost(W)(1 + j)4+k exp(−2εj).

The exponential series gives (1 +. Summing proves a hit budget with tail factor exp() and payment

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exp(av)γCP ballCost(W)
·ε−(4+k)(4 + k)! exp(ε)/(1 −exp(−ε)).

The argument bounds every finite label subset, so it proves the nonnegative countable summability as well. For a block parent of side L, the site decomposition into parent and residue gives. For an old support Y at scale, the coarse collapse estimate gives

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coverNumberB(halolB(parent(Y )))
≤preCost(l) + 10 preCost(l) coverNumberM(Y )/L,
preCost(l) = ballCost(l + 6).

Its extent estimate gives separation from the parent of any chosen anchor at most (l+20) coverNumber )). These two bounds supply v,v and W in the general principle. The connected traversal and coarse-collapse arguments above prove these covering and extent bounds; the displayed summation then supplies the hit budget. Inspecting a support adds at most four fine-site steps, hence at most five parent steps, so its enlarged support is included after increasing l by five.

The reserve inequalities have the required quantifier order. For old exponent, choose L strictly above 10 preCost(). Then 10 preCost(, before choosing x. For path supports the coefficient is instead 10preCost()/(LB); at fixed L it tends to zero as any suffices eventually. The same delay ensures the old fine radius radiusB( Fixed powers of these radii and the monitor are subpower in betaF, whereas exp( till()) and exp( 0, beat every inverse power. These numerical facts justify the strict-exponent absorptions used above, without changing L after the delay.

Support tariffs and path-tail conversion

For nonnegative coefficient envelopes, supports and integer extra charges, the sales condition means that at every coarse site x,

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X
i:x∈Xi
wi exp(2εei)
exp(a coverNumberB(Xi)) ≤γ,

with the corresponding nonnegative series summable. Disjoint alphabets add their bounds. A finite additional orientation index costs the sum of the orientation bounds; rotating or translating supports preserves a bound since the site map is bijective and does not increase their covering number. If is nonempty and0, each follows by reading the inequality at one site in

To transfer an old anchored bound at scale M, reserve its extra tarifffirst: set exp(2 coverNumber)). Then

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vi exp((δ −2ε) coverNumberM(Yi))
= wi exp(δ coverNumberM(Yi)).

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The old anchored bound is therefore unchanged after reducing its support exponent by 2ε. If
the new Xi lies in the l-halo enlargement of the parent image of Yi, M ≤2B, and the geometric
reserve inequality holds, the coarse support estimate applies to v.
It allows a new coefficient
zi ≤Cwi(1 + covM(Yi))k and pays its new 2ε tariff. Its resulting budget is the explicit footprint
payment with fine-child factor L4, extentCost(l, B), degree k, original budget γ, support exponent
a and geometric preCost(l). The anchor-to-hit summation and the geometric coarsening proof are
given in (NF9)–(NF10) below, with these parameters.
For rooted path bundles the same transfer starts from a true length budget P
iwi

exp( length Set exp(2 length), leaving width for the geometric conversion. The bundle support estimate then supplies sales for any(1 + length on an allowed enlarged bundle footprint. Unlike an anchored activity family, its root varies over every fine site; the payment’s originFee and factors account for that census.

An inspected order-r atom footprint lies inside the (l+5)-halo enlargement of its actual path bundle. Applying the path transfer to its magnitude and exponential length budget gives the atom payment. Its degree-k polynomial mark is retained in the payment. For paired tails the mask halo adds only two further coarse halos: enlargement) is contained in the ( halo enlargement of the two-path bundle, by the parent-distance inequality. This supplies the tail payment with

The tail’s small factor can be computed before support conversion. Write x,y for the two actual slot lengths. A nonzero omitted pair has x or y. The pair-price bound and half of the exponential length reserve give

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pairPrice · exp((σ/2)(x + y))
≤2 exp(−σR/(2d))
|ux| exp(σx)|uy| exp(σy).

Sum using the product of the two original path budgets, each at most. Thus the half-width budget is at most 2 exp( = till() this has every prescribed negative rate. The remaining width is positive under 8, leaving the geometric reserve needed for support conversion and the polynomial mark.

All footprint payments multiply their scalar envelope and original budget by fixed constants and a four-dimensional extent factor. If those envelopes have rates u,v and the extent has rate zero, payment has rate u+v. Consequently any strictly larger exponent absorbs the remaining fixed and subpower factors. This proves the atom and tail rate deductions used by the inherited six-family assembly, conditional only on the geometric footprint conversion.

Inherited coefficients and their enlarged supports

Fix halo l and positive with 4 below the incoming support exponent, and 8 below both the fine and coarse path widths. A sufficiently large block length L converts the incoming anchored support budget to the enlarged coarse footprints, leaving the prescribed support price and extra-charge tariff. The same L works for the regular and forced activity classes. This conversion uses the original incoming envelopes, before any new marking reserve is constructed.

For a regular occurrence, take its supplied nonnegative envelope. On its healthy support its absolute value is bounded by using the zeroth-order incoming bound; outside that condition the masked regular occurrence is zero. Its anchor remains the old regular anchor. For a forced coefficient use its incoming envelope without a health restriction. Choose an inventory point in each forced support as anchor; the incoming hit bound implies the weaker anchored bound at that point. Apply the same coarse support conversion separately to each family. Its geometric payment factor has rate zero, because its extent is a fixed multiple of B and L is fixed. The regular incoming envelope has rate11/10, so any price with10 is available eventually. The forced incoming bound can be chosen with rate v1, yielding a sales price for any fixed v.

For an order-k path atom multiplied by a scalar with), let D have rate u and its total atom budget have rate v. The healthy atom bound gives the pointwise envelope

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z(site,j) = D(x) fiery(x)kmagnitude(atomj).

Its rate before support conversion is uk/2+v. If this is strictly below a desired exponent r, choose an intermediate exponent s between them, convert the inspected supports, sum over the finite signed orientations, and absorb that fixed number into. This retains the atom’s actual path length in its extra charge.

The omitted path-pair tails have a stronger property: their exponential length budget pays any prescribed power after a fixed scalar multiplier D of any fixed rate. The support conversion uses 8 and the actual cutofftill(), not endpoint separation. Sum the finitely many plaquette types and signed orientations at exponent r1 and absorb their fixed count into exponent r. The detailed atom and tail support-conversion estimates are still the inputs to these two deductions.

The inherited logarithmic exponent has six disjoint families: regular occurrences, cubic atoms, quartic atoms, loop atoms, fine path tails, and coarse path tails. Its scalar phase is . The beta band and1 imply2 betaF; the coarse tail uses and obeys the same bound. The cubic and quartic multipliers therefore have rate 1, giving exponents 12 and 11. The loop multiplier has rate zero and uses its own incoming loop budget. The two tail families allow arbitrary negative powers.

Given2, choose. All six families have sales bounds eventually, with the incoming regular bound weakened to s. Their disjoint-union price is at most 6

eventually. The forced family retains its independent price for any r. This proves the inherited dictionary bound with the actual fine and coarse beta factors already included. Every footprint remains nonempty: regular and forced supports retain their anchor or inventory, atom

footprints retain the rooted path origin, and tail-pair footprints retain their designated root. These facts support the individual-letter smallness argument used in the optional dictionary assembly.

The non-warning primitive dictionary

Before taking connected components, keep three disjoint kinds of logarithmic letter: inherited fine-scale log letters, local short-correction letters, and Gaussian chain letters. The last kind has five families: the negative actual log-determinant series, the positive blank-reference log-determinant series, the mean-source series, the first linear mean correction, and the remaining quadratic chain correction. In the integrand their signs and factors are respectivelyArmA/2, + ArmA +ArmS/2, qfirst and chain. These signs belong to the exponent itself.

For an inherited log letter take its actual old exponent evaluated on the lifted fine field and multiply it by the enforcement profile of its own full footprint at parameter 2a. For an inherited forced activity use its actual K coefficient with the same footprint enforcement. These profiles lie in [0,1], so they cannot increase the supplied absolute bound. The full exponent farLog is the disjoint sum of these inherited exponents, the Short correction, and the five chain families. Each optional activity is exp(farLog)

Supports are assigned before evaluating any coefficient: inherited letters use their old support enlarged by the prescribed halo; short letters use their sleeves; the chain series use their receipt supports, and the first correction uses its town support. Their extra charges are the old extra charges, zero for short letters and first corrections, and the series index for the other chain letters. Every support is nonempty, including coefficients which happen to vanish. This keeps the support price independent of a field-dependent selection.

The numerical assembly has the following exact form. Let the Gaussian-chain estimates give envelope families with a common Sales bound) of rate1/2. Applying the same estimate to the blank geometry controls the positive reference determinant term with its own envelope, not by identifying it with the actual determinant. Let the local Short estimate have envelope), whose support sales bound) has rate1/2. For the latter count, the number of captions per coarse site is at most

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|Signed|spots(L) + 2itineraryDensity(L),

and a sleeve of halo l is contained in the radius-(l+1)B neighborhood of its caption. Its common sales price is bounded by

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ballCost((l + 1)B) · captionDensity · Z(x)
· exp(v ballCost(l + 1)).

The radius census has rate zero, so this retains rate The inherited log and forced activities have their own preassigned envelopes from the incoming representation. Combine the log envelope, the Short envelope, and the chain envelopes by disjoint union to obtain. Their sales prices add. Choose exponents 0. The inherited log budget can be taken as, whereas X+N has rate

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T(x) = betaF(x)s + X(x) + N(x)

has rate s. Eventually1 and 2. These thresholds are chosen uniformly before quantifying over the geometry and fields. The combined sales bound and nonempty support imply1 for every letter. The absolute logarithm is bounded by, and

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| exp(t) −1| ≤2b0iwhen|t| ≤b0i ≤1.

Thus bounds both the raw logarithm and its Mayer activity, with combined sales price at most. The inherited forced envelopes retain their separately prescribed

budget for any fixed u. A dummy geometry at the same x, cutoffand width is used only to read the common numerical envelopes; the bounds themselves apply to every admitted geometry, every forest, and every field. This completes the dictionary assembly conditional on its inherited, Short and chain estimates, which are unfolded next.

Local driving estimate for the nonlinear Gaussian floor

The local driving estimate supplies the quantitative bridge between the geometric predictor and the activity expansion. It must be applied to the specific retained and noise coordinates constructed above, on their declared healthy chart. The proof keeps the quadratic positive floor while assigning the cubic, quartic and path errors to the finite envelopes already selected. The remaining subsections convert those anchored envelopes into the support-hit bounds required for the marked integration.

We now supply the local estimate used in the earlier quadratic-plate absorption argument. Fix one real accepted sector and its forest. For a variation, let be its restriction to the active bonds. At a coarse site

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



1/2
1/2

X
dist(o,addr(e))≤3B
∥uhot(e)∥2

X
dist(o,root(p))≤Rmac+2
∥motion(u)p∥2


Io(u) =
,
Ro(u) =
.

Each individual term in either sum is bounded by its square root, because all summands are nonnegative. Suppose a gauge transformation makes all background links within distance 2 of have chord at most0. The accepted local stencil estimate then says that the flat row and the background-oriented row, applied to Ad at a rod rooted within most

This uses only the background links on that local box. Adjoint action preserves the color norm. The rod is selected because an active bond is separated from the forbidden seeds by the prescribed buffer; the triangle inequality places its entire stencil inside the clean region. Gauge covariance identifies the oriented row with the corresponding measured motion component. Its norm is at most). Consequently the flat row is bounded by

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Eo(u) = Ro(u) + 2CcG(B + 1)h Io(u).
(NF1)

Here is how the local Poincaré estimate turns (NF1) into control of a bond. Write for the logarithmic color variation of Ad On every forest trunk is zero, hence its collected logarithmic variation is zero. The forest link estimate therefore bounds the variation at an active bond, with = addr(

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∥uhot(e)∥≤K
X
X
α∈PlaqAxes
∥curl w(x, α)∥+ ∥bill w(o, eaxis)∥.
x∈box(Rmac)

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The low face row in (NF1) is √qdir curl w, where qdir > 0 is the direct quadratic weight. Its norm

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is at most Eo(u), so each face norm in this sum is at most q−1/2
dir
Eo(u). The pin row is −
√
k bill w,